Subgroups — C₈×Q₈×#261 #16797447

Order 3328.

1070 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₁₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×#261	C₂	1	1	yes	maximal, characteristic
3	#13130629	C₂	1	3	yes	maximal
4	C₈×C₄⋊#261	C₂	1	3	yes	maximal
5	C₄⋊C₈×#261	C₂	1	3	yes	maximal
6	C₄×C₈×#261	C₂	1	3	yes	maximal
7	C₈×Q₈×D₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×#261	C₄	1	2	yes	—
9	C₄×Q₈×D₁₃	C₂²	1	1	yes	characteristic
10	Q₈×#1057	C₄	1	2	yes	complemented
11	C₂×C₈×#261	C₂²	1	3	yes	—
12	C₂×C₈×#261	C₂²	1	3	yes	—
13	C₂×C₈×#261	C₂²	1	3	yes	—
14	C₂×C₈×#261	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×#261	C₄	1	6	yes	—
16	#11719074	C₄	1	6	yes	complemented
17	C₄×C₄⋊#261	C₂²	1	3	yes	—
18	C₄²×#261	C₂²	1	3	yes	—
19	D₁₃⋊(C₄⋊C₈)	C₂²	1	3	yes	—
20	C₄×D₁₃⋊C₈	C₂²	1	3	yes	—
21	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic, complemented
22	Q₈×#1055	C₄	1	1	yes	characteristic, complemented
23	D₁₃×C₄⋊C₈	C₂²	1	3	yes	—
24	#11718491	C₄	1	3	yes	complemented
25	C₄×C₈×D₁₃	C₂²	1	3	yes	—
26	C₈×Dic₂₆	C₄	1	3	yes	complemented
27	(C₂×#261)⋊C₈	C₂²	1	6	yes	—
28	C₂×Q₈×D₁₃	C₂×C₄	1	1	yes	characteristic
29	Q₈×#261	C₈	1	16	yes	direct product factor
30	C₂×C₄⋊#261	C₂×C₄	1	6	yes	—
31	C₂×C₄×#261	C₂×C₄	1	6	yes	—
32	C₂×C₄×#261	C₂×C₄	1	6	yes	—
33	C₂×C₄×#261	C₂³	1	3	yes	—
34	C₂×C₄×#261	C₂³	1	1	yes	characteristic
35	C₂×D₁₃⋊C₈	C₂³	1	3	yes	—
36	C₂×D₁₃⋊C₈	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	Q₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₁₃	C₂³	1	3	yes	—
40	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×D₁₃	C₂×C₄	1	3	yes	—
42	#88141	C₂×C₄	1	3	yes	—
43	C₄²×D₁₃	C₂³	1	3	yes	—
44	C₄×Dic₂₆	C₂×C₄	1	3	yes	—
45	(C₂×#261)⋊C₄	C₂×C₄	1	6	yes	—
46	#88112	C₂×C₄	1	6	yes	—
47	C₄⋊#1057	C₂×C₄	1	6	yes	—
48	C₄×#1057	C₂×C₄	1	6	yes	—
49	C₄×#1057	C₂×C₄	1	6	yes	—
50	C₈×#261	—	2	6	no	—
51	C₈×#261	—	2	6	no	—
52	C₈×#261	—	2	6	no	—
53	C₈×#261	Q₈	1	4	yes	direct product factor
54	C₁₃×C₄⋊C₈	C₂×C₄	1	3	yes	—
55	C₄×C₈×C₁₃	C₂×C₄	1	3	yes	—
56	#88054	C₂×C₄	1	6	yes	—
57	C₈×Dic₁₃	C₂×C₄	1	3	yes	—
58	C₈×Dic₁₃	C₂×C₄	1	3	yes	—
59	C₄⋊#1055	C₂×C₄	1	3	yes	—
60	C₄×#1055	C₂×C₄	1	3	yes	—
61	C₄×C₈×Q₈	—	13	1	no	maximal, Sylow
62	C₂²×#261	C₂²×C₄	1	6	yes	—
63	C₂²×#261	C₂²×C₄	1	2	yes	—
64	C₂×Q₈×C₁₃	C₄²	1	1	yes	characteristic
65	Q₈×D₁₃	C₂×C₈	1	8	yes	—
66	C₂×C₄×D₁₃	C₂²×C₄	1	3	yes	—
67	C₂×C₄×D₁₃	C₂²×C₄	1	3	yes	—
68	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
69	C₂×Dic₂₆	C₄²	1	3	yes	—
70	C₂×#1057	C₂²×C₄	1	6	yes	—
71	C₂×#1057	C₂²×C₄	1	2	yes	—
72	C₄⋊#261	C₂×C₈	1	24	yes	—
73	C₄×#261	—	2	12	no	—
74	C₄×#261	—	2	3	no	—
75	C₄×#261	—	2	3	no	—
76	C₄×#261	C₂×C₈	1	24	yes	—
77	C₄×#261	C₂×Q₈	1	2	yes	—
78	C₄×#261	#41	1	2	yes	—
79	D₁₃⋊C₈	—	2	3	no	—
80	D₁₃⋊C₈	—	2	3	no	—
81	D₁₃⋊C₈	C₂×Q₈	1	2	yes	—
82	D₁₃⋊C₈	#41	1	2	yes	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₁₃×C₄⋊C₄	C₄²	1	3	yes	—
86	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
87	C₄⋊Dic₁₃	C₄²	1	3	yes	—
88	#6101	C₄²	1	6	yes	—
89	C₄×Dic₁₃	C₂²×C₄	1	3	yes	—
90	C₄×Dic₁₃	C₂²×C₄	1	3	yes	—
91	C₄×Dic₁₃	C₂²×C₄	1	3	yes	—
92	C₂×#1055	C₂²×C₄	1	3	yes	—
93	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
94	C₈×D₁₃	—	2	3	no	—
95	C₈×D₁₃	—	2	3	no	—
96	C₈×D₁₃	C₂×Q₈	1	2	yes	—
97	C₈×D₁₃	#41	1	2	yes	—
98	C₂×C₈×Q₈	—	13	1	no	—
99	C₂×C₈×Q₈	—	13	1	no	—
100	C₄²×Q₈	—	13	1	no	—
101	C₈×C₄⋊C₄	—	13	3	no	—
102	C₈×C₄⋊C₄	—	13	3	no	—
103	C₄×C₄⋊C₈	—	13	3	no	—
104	C₄²×C₈	—	13	3	no	—
105	C₂²×D₁₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×#261	—	2	6	no	—
107	C₂×#261	—	2	6	no	—
108	C₂×#261	C₄×Q₈	1	4	yes	—
109	C₂×#261	C₂²×C₈	1	12	yes	—
110	C₂×#261	C₂²×C₈	1	4	yes	—
111	C₂×#261	#131	1	4	yes	—
112	Q₈×C₁₃	C₄×C₈	1	4	yes	complemented
113	C₂×C₄×C₁₃	C₂×C₄²	1	3	yes	—
114	C₂×C₄×C₁₃	C₂×C₄²	1	3	yes	—
115	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×Dic₁₃	C₂×C₄²	1	3	yes	—
117	C₂×Dic₁₃	C₂×C₄²	1	3	yes	—
118	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₄×D₁₃	—	2	6	no	—
120	C₄×D₁₃	C₂²×C₈	1	12	yes	—
121	C₄×D₁₃	C₂²×Q₈	1	1	yes	characteristic
122	C₄×D₁₃	C₂×#41	1	1	yes	characteristic
123	C₄×D₁₃	C₂×#41	1	1	yes	characteristic
124	C₄×D₁₃	C₂×#41	1	1	yes	characteristic
125	Dic₂₆	C₄×C₈	1	12	yes	complemented
126	#1057	—	2	6	no	—
127	#1057	C₄×Q₈	1	4	yes	complemented
128	C₈×C₁₃	—	2	3	no	—
129	C₈×C₁₃	C₄×Q₈	1	2	yes	complemented
130	#1055	—	2	3	no	—
131	#1055	C₄×Q₈	1	2	yes	complemented
132	C₂×C₄×Q₈	—	13	2	no	—
133	C₂×C₄×Q₈	—	13	1	no	—
134	C₈×Q₈	—	13	6	no	—
135	C₈×Q₈	—	13	2	no	—
136	C₈×Q₈	—	13	1	no	—
137	C₈×Q₈	—	13	3	no	—
138	C₈×Q₈	—	13	3	no	—
139	C₈×Q₈	#261	1	1	yes	characteristic, direct product factor
140	C₂×C₄⋊C₈	—	13	3	no	—
141	C₂×C₄⋊C₈	—	13	3	no	—
142	C₂×C₄×C₈	—	13	3	no	—
143	C₂×C₄×C₈	—	13	1	no	—
144	C₂×C₄×C₈	—	13	3	no	—
145	C₂×C₄×C₈	—	13	3	no	—
146	C₂×C₄×C₈	—	13	3	no	—
147	C₂×C₄×C₈	—	13	3	no	—
148	C₄×C₄⋊C₄	—	13	3	no	—
149	C₄×C₄⋊C₄	—	13	6	no	—
150	C₄³	—	13	3	no	—
151	#336	—	13	6	no	—
152	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
153	C₂×D₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
154	C₂×D₁₃	C₄×#41	1	1	yes	characteristic
155	C₂×D₁₃	C₂³×C₈	1	2	yes	—
156	C₂×D₁₃	C₂×#131	1	2	yes	—
157	#261	—	2	12	no	—
158	#261	C₈×Q₈	1	8	yes	direct product factor
159	C₄×C₁₃	—	2	3	no	—
160	C₄×C₁₃	C₂×C₄×C₈	1	6	yes	—
161	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
162	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
163	Dic₁₃	—	2	3	no	—
164	Dic₁₃	C₂×C₄×C₈	1	6	yes	—
165	Dic₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
166	Dic₁₃	C₄×#41	1	1	yes	characteristic
167	C₂²×Q₈	—	13	1	no	—
168	C₂²×C₈	—	13	1	no	—
169	C₂²×C₈	—	13	1	no	—
170	C₂²×C₈	—	13	3	no	—
171	C₂²×C₈	—	13	3	no	—
172	C₄×Q₈	—	13	16	no	—
173	C₄×Q₈	—	13	1	no	—
174	C₄×Q₈	—	13	3	no	—
175	C₄×Q₈	—	13	3	no	—
176	C₄×Q₈	C₂×#261	1	1	yes	characteristic
177	C₂×C₄⋊C₄	—	13	6	no	—
178	C₂×C₄⋊C₄	—	13	3	no	—
179	C₂×C₄²	—	13	6	no	—
180	C₂×C₄²	—	13	1	no	—
181	C₂×C₄²	—	13	3	no	—
182	C₂×C₄²	—	13	3	no	—
183	C₂×C₄²	—	13	6	no	—
184	C₄⋊C₈	—	13	6	no	—
185	C₄⋊C₈	—	13	6	no	—
186	C₄⋊C₈	—	13	6	no	—
187	C₄⋊C₈	—	13	3	no	—
188	C₄⋊C₈	C₂×#261	1	3	yes	—
189	C₄×C₈	—	13	4	no	—
190	C₄×C₈	—	13	6	no	—
191	C₄×C₈	—	13	6	no	—
192	C₄×C₈	—	13	3	no	—
193	C₄×C₈	—	13	3	no	—
194	C₄×C₈	—	13	3	no	—
195	C₄×C₈	—	26	6	no	—
196	C₄×C₈	—	26	6	no	—
197	C₄×C₈	—	26	6	no	—
198	C₄×C₈	C₂×#261	1	3	yes	—
199	#95	—	13	6	no	—
200	C₂×C₁₃	C₄²×Q₈	1	1	yes	characteristic
201	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
202	C₂×C₁₃	C₄×#131	1	1	yes	characteristic
203	D₁₃	C₂×C₈×Q₈	1	2	yes	—
204	D₁₃	C₈×#41	1	2	yes	—
205	C₂×Q₈	—	13	8	no	—
206	C₂×Q₈	—	13	3	no	—
207	C₂×Q₈	C₄×#261	1	1	yes	characteristic
208	C₂²×C₄	—	13	2	no	—
209	C₂²×C₄	—	13	3	no	—
210	C₂²×C₄	—	13	1	no	—
211	C₂²×C₄	—	13	6	no	—
212	C₂²×C₄	—	13	3	no	—
213	C₂×C₈	—	13	2	no	—
214	C₂×C₈	—	13	2	no	—
215	C₂×C₈	—	13	2	no	—
216	C₂×C₈	—	13	1	no	—
217	C₂×C₈	—	13	2	no	—
218	C₂×C₈	—	13	2	no	—
219	C₂×C₈	—	13	6	no	—
220	C₂×C₈	—	13	3	no	—
221	C₂×C₈	—	26	3	no	—
222	C₂×C₈	—	26	3	no	—
223	C₂×C₈	—	26	3	no	—
224	C₂×C₈	—	26	3	no	—
225	C₂×C₈	C₂²×#261	1	3	yes	—
226	C₂×C₈	C₂²×#261	1	1	yes	characteristic
227	C₄⋊C₄	—	13	24	no	—
228	C₄⋊C₄	—	13	6	no	—
229	C₄⋊C₄	—	13	3	no	—
230	C₄⋊C₄	C₄×#261	1	3	yes	—
231	C₄²	—	13	2	no	—
232	C₄²	—	13	2	no	—
233	C₄²	—	13	24	no	—
234	C₄²	—	13	3	no	—
235	C₄²	—	13	3	no	—
236	C₄²	—	13	3	no	—
237	C₄²	—	26	3	no	—
238	C₄²	—	26	3	no	—
239	C₄²	—	26	12	no	—
240	C₄²	C₂²×#261	1	3	yes	—
241	C₁₃	C₄×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
242	C₂³	—	13	1	no	—
243	Q₈	—	13	12	no	—
244	Q₈	C₈×#261	1	4	yes	direct product factor
245	C₂×C₄	—	13	4	no	—
246	C₂×C₄	—	13	4	no	—
247	C₂×C₄	—	13	4	no	—
248	C₂×C₄	—	13	1	no	—
249	C₂×C₄	—	13	3	no	—
250	C₂×C₄	—	13	12	no	—
251	C₂×C₄	—	13	1	no	—
252	C₂×C₄	—	13	1	no	—
253	C₂×C₄	—	13	1	no	—
254	C₂×C₄	—	13	1	no	—
255	C₂×C₄	—	13	12	no	—
256	C₂×C₄	—	13	3	no	—
257	C₂×C₄	—	26	6	no	—
258	C₂×C₄	—	26	6	no	—
259	C₂×C₄	—	26	6	no	—
260	C₂×C₄	C₂×C₄×#261	1	3	yes	—
261	C₂×C₄	C₂×C₄×#261	1	3	yes	—
262	C₂×C₄	C₂³×#261	1	1	yes	characteristic
263	C₈	—	2	3	no	—
264	C₈	—	13	4	no	—
265	C₈	—	13	2	no	—
266	C₈	—	26	6	no	—
267	C₈	—	26	3	no	—
268	C₈	Q₈×#261	1	2	yes	direct product factor
269	C₂²	—	13	1	no	—
270	C₂²	—	13	1	no	—
271	C₂²	—	13	2	no	—
272	C₂²	—	13	2	no	—
273	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
274	C₄	—	2	3	no	—
275	C₄	—	13	8	no	—
276	C₄	—	13	1	no	—
277	C₄	—	13	1	no	—
278	C₄	—	13	6	no	—
279	C₄	—	26	12	no	—
280	C₄	—	26	3	no	—
281	C₄	C₂×C₈×#261	1	6	yes	—
282	C₄	C₂×Q₈×#261	1	1	yes	characteristic
283	C₄	#41×#261	1	1	yes	characteristic
284	C₂	—	13	2	no	—
285	C₂	—	13	2	no	—
286	C₂	C₄×Q₈×#261	1	1	yes	characteristic
287	C₂	C₂²×C₈×#261	1	1	yes	characteristic
288	C₂	#131×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×#261
commutator	C₂×C₁₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₈×Q₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂²×C₄²
prime radical	C₂²×D₁₃	C₂³×C₄
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	C₈×Q₈	#261
hypocenter	C₁₃	C₄×C₈×Q₈
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
10. Q₈×#1057	C₄	semidirect
16. #11719074	C₄	semidirect
21. C₈×Q₈×C₁₃	C₄	semidirect
22. Q₈×#1055	C₄	semidirect
24. #11718491	C₄	semidirect
26. C₈×Dic₂₆	C₄	semidirect
29. Q₈×#261	C₈	direct (N × H)
53. C₈×#261	Q₈	direct (N × H)
112. Q₈×C₁₃	C₄×C₈	semidirect
125. Dic₂₆	C₄×C₈	semidirect
127. #1057	C₄×Q₈	semidirect
129. C₈×C₁₃	C₄×Q₈	semidirect
131. #1055	C₄×Q₈	semidirect
139. C₈×Q₈	#261	direct (N × H)
158. #261	C₈×Q₈	direct (N × H)
241. C₁₃	C₄×C₈×Q₈	semidirect
244. Q₈	C₈×#261	direct (N × H)
268. C₈	Q₈×#261	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
Q₈×D₁₃⋊C₈	2
C₄×Q₈×#261	2
#13130629	2
C₈×C₄⋊#261	2
C₄⋊C₈×#261	2
C₄×C₈×#261	2
C₈×Q₈×D₁₃	2
C₄×C₈×Q₈	13

Subgroups — C8×Q8×#261 #16797447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×Q₈×#261 #16797447